Find $G_a$ in the following case ${a_n}={1\over{(n-1)(n+1)}}$ for $n\ge 2$

Question

We briefly covered generating functions in class and most of the situations we covered we were given a recurrence to find a generating function for. I haven't gotten very far but I do believe $$G_a(x) = \sum_{n=2}^\infty({x^n\over(n-1)(n+1)})$$ and i think i will have to differentiate at some point but I am lost

Answer 1

Hint: If we rewrite the $n$-th term as $$\frac{1}{2}\cdot \frac{x^n}{n-1}-\frac{1}{2}\cdot \frac{x^n}{n+1},$$ things may look more familiar.